Hard Optimization problem

ynath23

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Mar 24, 2013
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An electric utility is required to run a cable from a transformer station on the shore of a lake to an island. The island is 5 km from the shore and the station is 12 km down the shoreline from a point opposite the island. It costs $5000/km to run the cable on land and $7000/km underwater. Find the path the cable should take for a minimum cost of installation.
 
An electric utility is required to run a cable from a transformer station on the shore of a lake to an island. The island is 5 km from the shore and the station is 12 km down the shoreline from a point opposite the island. It costs $5000/km to run the cable on land and $7000/km underwater. Find the path the cable should take for a minimum cost of installation.
We need to see your work - else we don't know where you are getting stuck.

Assuming you need help setting it up, the first thing to do is draw a picture. I would put the origin at the point on the shore opposite the island. Island would be at (0,5 km), and the power plant at (12 km,0). Set up your cost equation in that coordinate system.

If you still need help, show ys how far you have gotten.
 
Screen Shot 2013-03-24 at 4.48.23 PM.jpg

I have graphed the points and calculated the distance between the island and the station to be 13km. I'm confused as to the setup of the cost equations..
C=5000x+7000y
where x is km on land and y is km underwater
 
View attachment 2712

I have graphed the points and calculated the distance between the island and the station to be 13km. I'm confused as to the setup of the cost equations..
C=5000x+7000y
where x is km on land and y is km underwater
You have drawn two extreme cases: directly under water for 13 km @ $7000 per km, OR running 5 km under water (@ $7000 per km) plus 12 km on land (@ $5000 per km). Neither of those is likely to be the minimum cost. What if you come from the island to a point on shore which is somewhere between 0 and 12 km? Could you write an equation for the cost of such a path, and then minimize it?
 
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